Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342-avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x) turns out to be algebraic, proving the first nonmonotonic, longer-than-three instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.

Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig’s canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger. 1.

It is shown that if the automorphism group of a binary self-dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary self-dual code can have all its weights divisible by 4. The number of cyclic binary self-dual codes of length n is determined, and the shortest nontrivial code in this class is shown to have length 14. 1.

To every directed graph G one can associate a complex \Delta(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn , where it leads to studying some interesting representations of the symmetric group and corresponds (via Stanley-Reisner correspondence) to an interesting quotient ring.

We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie algebra G2. It is therefore related to G2 in the same way that the HOMFLY polynomial is related to An and the Kauffman polynomial is related to Bn, Cn, and Dn. We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions. This paper is divided into two parts. In the first part we derive from first principles some variants of the link invariant known as the Jones polynomial. In the second part we show that these invariants are the same known invariants constructed using rank 2 Lie algebras, and we discuss some of their properties. 1 Invariants of links and graphs The simplest known definition of the Jones polynomial is the Kauffman bracket [8], which in this paper will be denoted by 〈·〉A1 and will be called the A1 bracket. The A1 bracket is given by the following recursive rules: | 〉 A1 = −(q 1/2 +q-1/2) | 〉 A1 | 〉 A1 = −q 1/4 | 〉 A1 − q-1/4 | 〉 A1 The goal of this part of the paper is to derive definitions of the following three variants of the Jones polynomial: Theorem 1.1. There is an invariant for regular isotopy of projections of links and tangled trivalent graphs called 〈·〉G2 which is given by the following recursive rules:

Recently, Stanley [21] has defined a symmetric function generalization of the chromatic polynomial of a graph. Independently, Chung and Graham [4] have defined a digraph polynomial called the cover polynomial which is closely related to the chromatic polynomial of a graph (in fact, as

Abstract. A cellular graph is a graph whose edges can be partitioned into 4-cycles (called cells) so that each vertex is contained in at most two cells. We present a “Complementation Theorem ” for the number of matchings of certain subgraphs of cellular graphs. This generalizes the main result of [2]. As applications of the Complementation Theorem we obtain a new proof of Stanley’s multivariate version of the Aztec diamond theorem, a weighted generalization of a result of Knuth [7] concerning spanning trees of Aztec diamond graphs, a combinatorial proof of Yang’s enumeration [17] of matchings of fortress graphs and direct proofs for certain identities of Jockusch and Propp [6]. 1.

We use the Hopf algebra structure of the algebra of symmetric functions to study the Adams operators of the complex representation rings of symmetric groups, and we give new proofs of all of Littlewood's formulas for inner plethysm. We also study the Adams operations for orthogonal and symplectic group characters. 1 Background Our notations for symmetric functions (of an infinite, countable set of indeterminates X = fx 1 ; x 2 ; : : :g) will be the same as in [Mcd], except for the following minor changes. We denote by Sym(X) := n Sym (X) the Z-algebra of symmetric functions over X , graded by total degree. The standard scalar product on Sym(X) (for which the Schur functions form an orthonormal basis) is denoted by (\Delta ; \Delta) rather than by h\Delta ; \Deltai, the symbol h\Deltai being reserved for an other use. Also, for a symmetric function F , we denote by D F instead of D(F ) the adjoint of the linear operator G 7! FG. The generating series for complete and elementary functions are oe z (X) = h n (X) = (1 \Gamma zx i ) z (X) = e n (X) = (1 + zx i ) : A partition can be described by the sequence of its parts, arranged in decreasing order or by the multiplicities of the parts. If ff = (ff 1 ; ff 2 ; : : :) = (1 : : :), then we call the sequence a := (a 1 ; a 2 ; : : :) the cycle type of ff.

this paper we explore an approach which enables all of the problems listed above to be solved at a single stroke: use Lisp as the source language for the numeric and graphical code! This is not a new idea --- it was tried at MIT and UCB in the 1970's. While these experiments were modestly successful, the particular systems are obsolete. Fortunately, some of those ideas used in Maclisp [37], NIL [38] and Franz Lisp [20] were incorporated in the subsequent standardization of Common Lisp (CL) [35]. In this new setting it is appropriate to re-examine the theoretical and practical implications of writing numeric code in Lisp. The popular conceptions of Lisp's inefficiency for numerics have been based on rumor, supposition, and experience with early and (in fact) inefficient implementations. It is certainly possible to continue to write inefficient programs: As one example of the results of de-emphasizing numerics in the design, consider the situation of the basic arithmetic operators. The definitions of these functions require that they are generic, (e.g. "+" must be able to add any combination of several precisions of floats, arbitrary-precision integers, rational numbers, and complexes), The very simple way of implementing this arithmetic -- by subroutine calls -- is also very inefficient. Even with appropriate declarations to enable more specific treatment of numeric types, compilers are free to ignore declarations and such implementations naturally do not accommodate the needs of intensive number-crunching. (See the appendix for further discussion of declarations). Be this as it may, the situation with respect to Lisp has changed for the better in recent years. With the advent of ANSI standard Common Lisp, several active vendors of implementations and one active universi...

We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results; e.g., non-holonomicity of the logarithmic sequence is extended to rational functions involving log n. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at odd integers is not holonomic.